58 ideas
11912 | Substantive metaphysics says what a property is, not what a predicate means [Molnar] |
11920 | A real definition gives all the properties that constitute an identity [Molnar] |
13520 | A 'tautology' must include connectives [Wolf,RS] |
13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS] |
13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS] |
13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS] |
13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS] |
13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS] |
13531 | Model theory reveals the structures of mathematics [Wolf,RS] |
13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS] |
13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS] |
13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
8628 | I hold that algebra and number are developments of logic [Jevons] |
11919 | Ontological dependence rests on essential connection, not necessary connection [Molnar] |
11929 | The three categories in ontology are objects, properties and relations [Molnar] |
11927 | Reflexive relations are syntactically polyadic but ontologically monadic [Molnar] |
11915 | If atomism is true, then all properties derive from ultimate properties [Molnar] |
11916 | 'Being physical' is a second-order property [Molnar] |
11956 | 'Categorical properties' are those which are not powers [Molnar] |
11928 | Are tropes transferable? If they are, that is a version of Platonism [Molnar] |
11933 | A power's type-identity is given by its definitive manifestation [Molnar] |
11932 | Powers have Directedness, Independence, Actuality, Intrinsicality and Objectivity [Molnar] |
11934 | The physical world has a feature very like mental intentionality [Molnar] |
11947 | Dispositions and external powers arise entirely from intrinsic powers in objects [Molnar] |
11953 | Some powers are ungrounded, and others rest on them, and are derivative [Molnar] |
11952 | The Standard Model suggest that particles are entirely dispositional, and hence are powers [Molnar] |
11943 | Dispositions can be causes, so they must be part of the actual world [Molnar] |
11939 | If powers only exist when actual, they seem to be nomadic, and indistinguishable from non-powers [Molnar] |
11914 | Platonic explanations of universals actually diminish our understanding [Molnar] |
11913 | For nominalists, predicate extensions are inexplicable facts [Molnar] |
11962 | Nominalists only accept first-order logic [Molnar] |
11917 | Structural properties are derivate properties [Molnar] |
11955 | There are no 'structural properties', as properties with parts [Molnar] |
11918 | The essence of a thing need not include everything that is necessarily true of it [Molnar] |
11963 | What is the truthmaker for a non-existent possible? [Molnar] |
11951 | Hume allows interpolation, even though it and extrapolation are not actually valid [Molnar] |
11936 | The two ways proposed to distinguish mind are intentionality or consciousness [Molnar] |
11935 | Physical powers like solubility and charge also have directedness [Molnar] |
11944 | Rule occasionalism says God's actions follow laws, not miracles [Molnar] |
11960 | Singular causation is prior to general causation; each aspirin produces the aspirin generalization [Molnar] |
11937 | We should analyse causation in terms of powers, not vice versa [Molnar] |
11954 | We should analyse causation in terms of powers [Molnar] |
11961 | Causal dependence explains counterfactual dependence, not vice versa [Molnar] |
11959 | Science works when we assume natural kinds have essences - because it is true [Molnar] |
9448 | Location in space and time are non-power properties [Molnar, by Mumford] |
11930 | One essential property of a muon doesn't entail the others [Molnar] |
11957 | It is contingent which kinds and powers exist in the world [Molnar] |
11921 | The laws of nature depend on the powers, not the other way round [Molnar] |
11931 | Energy fields are discontinuous at the very small [Molnar] |